Generalized Fermi-Dirac Distribution of Exclusive Fermions

TL;DR

This work derives a generalized Fermi-Dirac distribution for exclusive fermions, where double occupancy of a quantum state is forbidden, using the grand canonical ensemble. The exclusive distribution is $f(ε)=2/(e^{β(ε-μ)}+2)$ and leads to an equation of state with degeneracy pressure enhanced relative to ordinary fermions; the Fermi energy at $T=0$ is $E_F = (\hbar^2/2m)(6π^2 N/V)^{2/3}$, with $E_F^{exclusive}=2^{2/3}E_F^{free}$. At low temperature, Sommerfeld-like expansions yield a specific heat $c ≈ 5.55\,k_B^2 T / E_F$, and the magnetic responses preserve Pauli paramagnetism and Landau diamagnetism forms, giving $M$ and $χ_{dia}$ as in the standard case. The results imply a larger degeneracy pressure and a correspondingly inflated Chandrasekhar limit (about 1.6× higher), with potential relevance for doped semiconductors and magnetized dense astrophysical objects. Overall, the paper extends Fermi-Dirac statistics to exclusive occupancy and maps its thermodynamics and magnetization properties.

Abstract

A system of exclusive fermions occurs when two fermions of opposite spin are prohibited from occupying the same quantum level. We derive the distribution of exclusive fermions via the employment of the grand canonical ensemble. Salient features of its statistical properties, compared to the free electron gases, include: larger Fermi energy, higher degeneracy pressure, but the same Pauli paramagnetism and Landau diamagnetism. In particular, higher degeneracy pressure leads to an inflation of the Chandrasekhar limit to 1.6 times when applied to white dwarf stars and neutron stars.